Relationship Between the Interacting Boson Model and the Bohr Collective Hamiltonian

It is demonstrated that the Interacting Boson Model I (IBM), which is described by a Hamiltonian of six coupled harmonic oscillators, (one s-boson and 5 d-bosons), where the total number of bosons, N, is constant of the motion, can be transformed into a Bohr-Mottelson model (BMM) described by a Hamiltonian with the familiar five dimensional quadrupole oscillator degrees of freedom. The correspondence is one-to-one for a BMM acting in a sub-space of the full five-dimensional harmonic oscillator space. The proof depends on a well-known non-linear realization of the generators of SU(6) (which are the basic building blocks of the IBM) in terms of the five BMM bosons and the number N. The orthonormal basis vectors of the BMM are obtained. The relationship between the limiting symmetries of the IBM and the geometrically simple limits of the BMM is described with the aid of the concept of potential energy surface. The form of the BMM Hamiltonian is given for the adiabatic limit. Finally we show that previous work on this problem involving coherent states and the generator coordinate method is incomplete.